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Theorem cnvss 4777
Description: Subset theorem for converse. (Contributed by NM, 22-Mar-1998.)
Assertion
Ref Expression
cnvss (𝐴𝐵𝐴𝐵)

Proof of Theorem cnvss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3136 . . . 4 (𝐴𝐵 → (⟨𝑦, 𝑥⟩ ∈ 𝐴 → ⟨𝑦, 𝑥⟩ ∈ 𝐵))
2 df-br 3983 . . . 4 (𝑦𝐴𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐴)
3 df-br 3983 . . . 4 (𝑦𝐵𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
41, 2, 33imtr4g 204 . . 3 (𝐴𝐵 → (𝑦𝐴𝑥𝑦𝐵𝑥))
54ssopab2dv 4256 . 2 (𝐴𝐵 → {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
6 df-cnv 4612 . 2 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
7 df-cnv 4612 . 2 𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}
85, 6, 73sstr4g 3185 1 (𝐴𝐵𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2136  wss 3116  cop 3579   class class class wbr 3982  {copab 4042  ccnv 4603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147
This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-in 3122  df-ss 3129  df-br 3983  df-opab 4044  df-cnv 4612
This theorem is referenced by:  cnveq  4778  rnss  4834  relcnvtr  5123  funss  5207  funcnvuni  5257  funres11  5260  funcnvres  5261  foimacnv  5450  tposss  6214  structcnvcnv  12410  pw1nct  13883
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